Quantum chaos and random matrix theories

Abstract

In this review paper we discuss some recent advances in understanding the dynamical localization and dynamical tunneling effects in quantal Hamiltonian mixed-type systems (which are generic), exhibiting regular motion on invariant tori for some initial conditions and chaotic motion for the complementary initial conditions in the classical phase space. In particular, we look at the level spacing distribution. In the asymptotic regime of the sufficiently deep semiclassical limit (sufficiently small effective Planck constant) the Berry-Robnik (1984) picture applies, which is very well established. We present a new quasi-universal semiempirical theory of the level spacing distribution in a regime away from the Berry-Robnik regime (the near semiclassical limit), by describing both the dynamical localization effects of chaotic eigenstates, and the tunneling effects which couple regular and chaotic eigenstates. The theory works extremely well in the 2D mixed type billiard system introduced by Robnik (1983) and is tested also in other systems (mushroom billiard and Prosen billiard).